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Chapter 2: Problem 49

Determine whether each relation defines \(y\) as a function of \(x\). \(y=-6 x\)

Short Answer

Expert verified
Yes, y=-6x defines y as a function of x.

Step by step solution

01

Determine the Type of Equation

Observe that the given relation is a linear equation of the form y = mx, where m is a constant. In this case, m equals -6.
02

Look for Multiple Outputs

Recall that for a relation to be a function, each input value (x) must correspond to exactly one output value (y). In the equation y = -6x, for each value of x, there is exactly one value of y.
03

Check Function Definition

Verify if there are any instances where one x-value could produce multiple y-values. In the given equation, for each unique input x, the output y is uniquely determined as y = -6x. Thus, no x-value produces more than one unique y-value.
04

Conclusion on Being a Function

Since each input value x generates a unique corresponding output value y, the relation y = -6xdefines y as a function of x.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equation

Let's begin by understanding what a linear equation is.
A linear equation is a type of equation where the highest power of the variable is 1.
It can be written in the form \(y = mx + b\), where \(m\) and \(b\) are constants.
In the case of the provided exercise, the linear equation is given as \(y = -6x\), which simplifies to \(y = mx\) with \(m = -6\) and \(b = 0\).

Some key points about linear equations:
  • They graph as straight lines.
  • There is a constant rate of change.
  • The slope of the line is determined by the coefficient \(m\).
Note that in the equation \(y = -6x\), the slope is -6, meaning for every unit increase in \(x\), \(y\) decreases by 6 units.
Unique Output
Now, let's delve into what it means for a function to have a unique output.
In mathematics, a function is a relation between sets that associates each element of a set (input) with exactly one element of another set (output).

For example, in our equation \(y = -6x\):
  • If \(x = 1\), then \(y = -6(1) = -6\).
  • If \(x = 2\), then \(y = -6(2) = -12\).
Notice that for each input value of \(x\), there is only one output value of \(y\).
This property of having a unique output ensures that the relation is indeed a function.

In simpler terms, no two different outputs \(y\) are produced for the same input \(x\) in the given linear equation.
Function Definition
Understanding the definition of a function is crucial for this exercise.
In basic terms, a function is a mathematical relation where each element from the input set corresponds to exactly one element in the output set.
This can be written as \(f(x)\rightarrow y\).
To satisfy the function definition:
  • Each input has a single, unique output.
  • An input \(x\) cannot produce multiple outputs \(y\).
In our given equation \(y = -6x\), we see that for every input \(x\), there is one and only one corresponding output \(y\).

Let's verify the function for some values:
  • For \(x = 0\), \(y = -6(0) = 0\).
  • For \(x = 5\), \(y = -6(5) = -30\).
This verification shows that there is a unique output \(y\) for each input \(x\), confirming that our equation defines \(y\) as a function of \(x\).

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Most popular questions from this chapter

Determine whether each pair of lines is parallel, perpendicular, or neither. \(x=6\) and \(6-x=8\)

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